## Optimal $$L^p$$-$$L^q$$-type decay rates of solutions to the three-dimensional nonisentropic compressible Euler equations with relaxation.(English)Zbl 1481.35069

Summary: In this paper, we consider the three-dimensional Cauchy problem of the nonisentropic compressible Euler equations with relaxation. Following the method of Y. Wu et al. [ibid. 2021, Article ID 5512285, 13 p. (2021; Zbl 07425109)], we show the existence and uniqueness of the global small $$H^k$$ ($$k \geqslant 3$$) solution only under the condition of smallness of the $$H^3$$ norm of the initial data. Moreover, we use a pure energy method with a time-weighted argument to prove the optimal $$L^p$$-$$L^q$$ ($$1 \leqslant p \leqslant 2$$, $$2 \leqslant q \leqslant \infty$$)-type decay rates of the solution and its higher-order derivatives.

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35B45 A priori estimates in context of PDEs 35Q31 Euler equations

### Keywords:

energy method; Cauchy problem; small initial data

Zbl 07425109
Full Text:

### References:

 [1] Li, T.; Zhao, K., Analysis of non-isentropic compressible Euler equations with relaxation, Journal of Differential Equations, 259, 11, 6338-6367 (2015) · Zbl 1332.35291 [2] Hsiao, L.; Luo, T., Nonlinear diffusive phenomena of solutions for the system of compressible adiabatic flow through porous media, Journal of Differential Equations, 125, 2, 329-365 (1996) · Zbl 0859.76067 [3] Hsiao, L.; Pan, R., Initial boundary value problem for the system of compressible adiabatic flow through porous media, Journal of Differential Equations, 159, 1, 280-305 (1999) · Zbl 0942.35138 [4] Marcati, P.; Pan, R., On the diffusive profiles for the system of compressible adiabatic flow through porous media, SIAM Journal on Mathematical Analysis, 33, 4, 790-826 (2001) · Zbl 0999.35077 [5] Dafermos, C. M.; Pan, R., Global BV solutions for the p-system with frictional damping, SIAM Journal on Mathematical Analysis, 41, 3, 1190-1205 (2009) · Zbl 1194.35255 [6] Hsiao, L.; Luo, T.; Yang, T., Global BV solutions of compressible Euler equations with spherical symmetry and damping, Journal of Differential Equations, 146, 1, 203-225 (1998) · Zbl 0916.35090 [7] Ding, X. X.; Chen, G. Q.; Luo, P. Z., Convergence of the fractional step Lax-Friedrichs scheme and Godunov scheme for the isentropic system of gas dynamics, Communications in Mathematical Physics, 121, 1, 63-84 (1989) · Zbl 0689.76022 [8] Geng, S.; Huang, F., L^1-convergence rates to the Barenblatt solution for the damped compressible Euler equations, Journal of Differential Equations, 266, 12, 7890-7908 (2019) · Zbl 1417.35101 [9] Huang, F.; Marcati, P.; Pan, R., Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum, Archive for Rational Mechanics and Analysis, 176, 1, 1-24 (2005) · Zbl 1064.76090 [10] Huang, F.; Pan, R., Convergence rate for compressible Euler equations with damping and vacuum, Archive for Rational Mechanics and Analysis, 166, 4, 359-376 (2003) · Zbl 1022.76042 [11] Huang, F.; Pan, R., Asymptotic behavior of the solutions to the damped compressible Euler equations with vacuum, Journal of Differential Equations, 220, 1, 207-233 (2006) · Zbl 1082.35031 [12] Huang, F.; Pan, R.; Wang, Z., L^1 convergence to the Barenblatt solution for compressible Euler equations with damping, Archive for Rational Mechanics and Analysis, 200, 2, 665-689 (2011) · Zbl 1229.35196 [13] Hsiao, L., Quasilinear Hyperbolic Systems and Dissipative Mechanisms (1997), River Edge, NJ, USA: World Scientific Publishing Co., Inc., River Edge, NJ, USA · Zbl 0911.35003 [14] Hsiao, L.; Liu, T.-P., Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Communications in Mathematical Physics, 143, 3, 599-605 (1992) · Zbl 0763.35058 [15] Pan, R.; Zhao, K., Initial boundary value problem for compressible Euler equations with damping, Indiana University Mathematics Journal, 57, 5, 2257-2282 (2008) · Zbl 1169.35040 [16] Pan, R.; Zhao, K., The 3D compressible Euler equations with damping in a bounded domain, Journal of Differential Equations, 246, 2, 581-596 (2009) · Zbl 1155.35066 [17] Hsiao, L.; Pan, R., The damped p-system with boundary effects, Nonlinear PDE’s, Dynamics and Continuum Physics (South Hadley, MA, 1998). Vol. 255 of Contemp. Math. Amer. Math. Soc. · Zbl 0941.00019 [18] Marcati, P.; Mei, M., Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping, Quarterly of Applied Mathematics, 58, 4, 763-784 (2000) · Zbl 1040.35044 [19] Nishihara, K.; Yang, T., Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping, Journal of Differential Equations, 156, 2, 439-458 (1999) · Zbl 0933.35121 [20] Nishihara, K., Convergence rates to nonlinear diffusion waves for solutions of system of hyperbolic conservation laws with damping, Journal of Differential Equations, 131, 2, 171-188 (1996) · Zbl 0866.35066 [21] Nishihara, K.; Wang, W.; Yang, T., L^p-convergence rate to nonlinear diffusion waves for p-system with damping, Journal of Differential Equations, 161, 1, 191-218 (2000) · Zbl 0946.35012 [22] Zhao, H., Convergence to strong nonlinear diffusion waves for solutions of p-system with damping, Journal of Differential Equations, 174, 1, 200-236 (2001) · Zbl 0990.35091 [23] Hsiao, L.; Serre, D., Global existence of solutions for the system of compressible adiabatic flow through porous media, SIAM Journal on Mathematical Analysis, 27, 1, 70-77 (1996) · Zbl 0849.35069 [24] Pan, R., Darcy’s law as long-time limit of adiabatic porous media flow, Journal of Differential Equations, 220, 1, 121-146 (2006) · Zbl 1079.76070 [25] Chen, Q.; Tan, Z., Time decay of solutions to the compressible Euler equations with damping, Kinetic & Related Models, 7, 4, 605-619 (2014) · Zbl 1318.35072 [26] Fang, D.; Xu, J., Existence and asymptotic behavior of C^1-solutions to the multi-dimensional compressible Euler equations with damping, Nonlinear Analysis, 70, 1, 244-261 (2009) · Zbl 1152.35432 [27] Liao, J.; Wang, W.; Yang, T., L^p convergence rates of planar waves for multi-dimensional Euler equations with damping, Journal of Differential Equations, 247, 1, 303-329 (2009) · Zbl 1170.35544 [28] Liu, Y.; Wang, W., Well-posedness of the IBVP for 2-D Euler equations with damping, Journal of Differential Equations, 245, 9, 2477-2503 (2008) · Zbl 1152.35070 [29] Sideris, T. C.; Thomases, B.; Wang, D., Long time behavior of solutions to the 3D compressible Euler equations with damping, Communications in Partial Differential Equations, 28, 3-4, 795-816 (2003) · Zbl 1048.35051 [30] Tan, Z.; Wang, Y., Global solution and large-time behavior of the 3D compressible Euler equations with damping, Journal of Differential Equations, 254, 4, 1686-1704 (2013) · Zbl 1259.35162 [31] Tan, Z.; Wu, G., Large time behavior of solutions for compressible Euler equations with damping in R3, Journal of Differential Equations, 252, 2, 1546-1561 (2012) · Zbl 1237.35131 [32] Wang, W.; Yang, T., The pointwise estimates of solutions for Euler equations with damping in multi-dimensions, Journal of Differential Equations, 173, 2, 410-450 (2001) · Zbl 0997.35039 [33] Wang, W.; Yang, T., Existence and stability of planar diffusion waves for 2-D Euler equations with damping, Journal of Differential Equations, 242, 1, 40-71 (2007) · Zbl 1147.35076 [34] Wei, R.; Li, Y.; Yao, Z.-A., Global existence and convergence rates of solutions for the compressible Euler equations with damping, Discrete & Continuous Dynamical Systems - B, 25, 8, 2949-2967 (2020) · Zbl 1440.35279 [35] Wu, Z.; Wang, W., Large time behavior and pointwise estimates for compressible Euler equations with damping, Science China Mathematics, 58, 7, 1397-1414 (2015) · Zbl 1334.35219 [36] Yang, X.; Wang, W., The suppressible property of the solution for three-dimensional Euler equations with damping, Nonlinear Analysis: Real World Applications, 8, 1, 53-61 (2007) · Zbl 1179.35187 [37] Wu, Y.; Wang, Y.; Shen, R., Global existence and long-time behavior of solutions to the full compressible Euler equations with damping and heat conduction in $$\mathbb{R}^3$$, Advances in Mathematical Physics, 2021 (2021) · Zbl 07425109 [38] Matsumura, A.; Nishida, T., The initial value problem for the equations of motion of viscous and heat-conductive gases, Journal of Mathematics of Kyoto University, 20, 1, 67-104 (1980) · Zbl 0429.76040 [39] Gao, J.; Li, M.; Yao, Z., Optimal decay of compressible Navier-Stokes equations with or without potential force (2021), http://arxiv.org/abs/2108.02453 [40] Nirenberg, L., On elliptic partial di_erential equations, The Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 13, 3, 115-162 (1959) · Zbl 0088.07601 [41] Ju, N., Existence and uniqueness of the solution to the dissipative 2D quasigeostrophic equations in the Sobolev space, Communications in Mathematical Physics, 251, 2, 365-376 (2004) · Zbl 1106.35061 [42] Wang, Y.; Liu, C.; Tan, Z., A generalized Poisson-Nernst-Planck-Navier-Stokes model on the fluid with the crowded charged particles: derivation and its well-posedness, SIAM Journal on Mathematical Analysis, 48, 5, 3191-3235 (2016) · Zbl 1416.35220 [43] Wang, Y.; Liu, C.; Tan, Z., Well-posedness on a new hydrodynamic model of the fluid with the dilute charged particles, Journal of Differential Equations, 262, 1, 68-115 (2017) · Zbl 1352.35127 [44] Stein, E. M., Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 (1970), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 0207.13501 [45] Sohinger, V.; Strain, R. M., The Boltzmann equation, Besov spaces, and optimal time decay rates in Rxn, Advances in Mathematics, 261, 274-332 (2014) · Zbl 1293.35195 [46] Guo, Y.; Wang, Y., Decay of dissipative equations and negative Sobolev spaces, Communications in Partial Differential Equations, 37, 12, 2165-2208 (2012) · Zbl 1258.35157
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.